Average Error: 38.7 → 0.0
Time: 3.6s
Precision: 64
Internal Precision: 128
\[\log \left(1 + x\right)\]
↓
\[\log_* (1 + x)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.7
Target0.3
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;1 + x = 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \log \left(1 + x\right)}{\left(1 + x\right) - 1}\\ \end{array}\]

Derivation

  1. Initial program 38.7

    \[\log \left(1 + x\right)\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\log_* (1 + x)}\]

  3. Final simplification0.0

    \[\leadsto \log_* (1 + x)\]

Reproduce

herbie shell --seed 2019016 +o rules:numerics
(FPCore (x)
  :name "ln(1 + x)"

  :herbie-target
  (if (== (+ 1 x) 1) x (/ (* x (log (+ 1 x))) (- (+ 1 x) 1)))

  (log (+ 1 x)))

Details

Time bar (total: 3.4s)Debug log

sample135.0ms

Algorithm
intervals
Results
385×(pre true 80)
128×(body nan 80)
81×(body real 80)
75×(body real 1280)
54×(body real 640)
32×(body real 320)
15×(body real 160)

simplify2.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
2.0ms
(log (+ 1 x))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize7.0ms

Local error

Found 0 expressions with local error:

rewrite0.0ms

Algorithm
rewrite-expression-head
Rules
Counts
0 → 0
Calls
0 calls:
Slowest

series0.0ms

Counts
0 → 0
Calls
0 calls:
Slowest

simplify0.0ms

Counts
0 → 0
Calls
0 calls:
Slowest

prune4.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

end0.0ms

sample3.2s

Algorithm
intervals
Results
10752×(pre true 80)
2849×(body real 80)
2752×(body nan 80)
2248×(body real 1280)
1693×(body real 640)
793×(body real 320)
417×(body real 160)